
Applying Lemma 7 from [1]}, with:
\(&\phi _1(x)=r_{0:t}(x) +c_{1:t-1}^\top x + \sum _{i=1}^{t-1} \lambda _i^\top g_i(x) + \tilde{c}_t^\top x +\lambda _t^\top \tilde{g}_t(x) \\&\phi _2(x)=r_{0:t}(x) +c_{1:t}^\top x + \sum _{i=1}^{t} \lambda _i^\top g_i(x) \quad \text{and} \\&\psi (x)=(c_{t}-\tilde{c}_t)^\top x + \lambda _t^\top \big (g_t(x)-\tilde{g}_t(x)\big ) \) 
